The work in this thesis is concerned with the solution of non-linear second order differential equations of relevance to bioelectrochemical systems, and the comparison of models thus derived with experimental results. A theoretical treatment of the cyclic voltammetric behaviour of homogeneous mediated redox enzyme systems is presented. This is supported by numerical simulation using the explicit finite difference method. The effects of Michaelis Menten enzyme kinetics, substrate concentration polarisation and enantiomeric forms of the substrate are considered. Case diagrams are derived showing the behaviour of the system. Theoretical and simulated results are shown to agree with experimental data for the glucose oxidase / glucose system mediated by ferrocene monocarboxylic acid. A theoretical treatment of the steady state behaviour of a type of immobilised enzyme electrode is presented. The enzyme is assumed to be immobilised uniformly in a layer at the electrode surface. The model considers the concentration profiles of both mediator and substrate within the enzyme layer. Michaelis Menten kinetics and electrode potential dependence are included. Situations are considered where the mediator is entrapped or bound within the film, and those where the mediator is present in the bulk solution in either its oxidised or reduced form. The theoretical model is supported by numerical simulation using the relaxation method; a steady state simulation technique.